Question:
Assertion (A): If $\vec{B}$ be the magnetic field at a small patch of vector area $d\vec{S}$, then the magnetic flux through the patch will be $\vec{B}\cdot d\vec{S}$.
Reason (R): The magnetic flux is a scalar quantity.
Assertion (A): If $\vec{B}$ be the magnetic field at a small patch of vector area $d\vec{S}$, then the magnetic flux through the patch will be $\vec{B}\cdot d\vec{S}$.
Reason (R): The magnetic flux is a scalar quantity.
Show Hint
Flux through a small patch is B.dS, a dot product, so it is a scalar. Both statements are true.
Updated On: Oct 21, 2026
- Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
- Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- Assertion (A) is true, but Reason (R) is false.
- Both Assertion (A) and Reason (R) are false.
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept:
Magnetic flux through a small patch is the amount of field passing through it. For a patch of vector area \(d\vec{S}\) (magnitude equal to the area, direction along the normal), the flux is \(d\phi = \vec{B}\cdot d\vec{S} = B\,dS\cos\theta\).
Step 2: Check the Assertion:
The definition of flux through a small patch is exactly the dot product \(\vec{B}\cdot d\vec{S}\).
Only the component of \(\vec{B}\) along the normal passes through the patch, and the dot product picks out this component.
So the Assertion is TRUE.
Step 3: Check the Reason:
A dot product of two vectors gives a number with no direction. So magnetic flux has magnitude only.
Flux has an algebraic sign, but no direction in space. So it is a scalar. The Reason is TRUE.
Step 4: Does the Reason explain the Assertion?
Flux is a scalar, so it must be built from two vectors by a scalar (dot) product. A cross product would give a vector.
This is why the flux is written as \(\vec{B}\cdot d\vec{S}\). The scalar nature of flux supports the dot-product form, so R can be taken as the explanation of A.
Step 5: Pick the option:
Both statements are true and R explains A, so (A) fits. Options (C) and (D) call a true statement false, so they are wrong.
Final Answer:
Both A and R are true and R explains A. This is option (A). \[ \boxed{\text{Option (A)}} \]
Magnetic flux through a small patch is the amount of field passing through it. For a patch of vector area \(d\vec{S}\) (magnitude equal to the area, direction along the normal), the flux is \(d\phi = \vec{B}\cdot d\vec{S} = B\,dS\cos\theta\).
Step 2: Check the Assertion:
The definition of flux through a small patch is exactly the dot product \(\vec{B}\cdot d\vec{S}\).
Only the component of \(\vec{B}\) along the normal passes through the patch, and the dot product picks out this component.
So the Assertion is TRUE.
Step 3: Check the Reason:
A dot product of two vectors gives a number with no direction. So magnetic flux has magnitude only.
Flux has an algebraic sign, but no direction in space. So it is a scalar. The Reason is TRUE.
Step 4: Does the Reason explain the Assertion?
Flux is a scalar, so it must be built from two vectors by a scalar (dot) product. A cross product would give a vector.
This is why the flux is written as \(\vec{B}\cdot d\vec{S}\). The scalar nature of flux supports the dot-product form, so R can be taken as the explanation of A.
Step 5: Pick the option:
Both statements are true and R explains A, so (A) fits. Options (C) and (D) call a true statement false, so they are wrong.
Final Answer:
Both A and R are true and R explains A. This is option (A). \[ \boxed{\text{Option (A)}} \]
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